Chapter 5 discusses the characteristics and implications of public goods, defined by nonexcludability and nonrivalry, along with private provision challenges such as free-riding and inefficiencies in decision-making. It explores voting mechanisms, personalized pricing, and mechanism design for optimal public good allocation, including the Lindahl and Clarke-Groves mechanisms. The chapter emphasizes the complexities involved in achieving efficient public good provision and the impact of income distribution on individual contributions.

1. Chapter 5
Public Goods

2. Reading
• Essential reading
– Hindriks, J and G.D. Myles Intermediate Public Economics. (Cambridge:
MIT Press, 2005) Chapter 5.
• Further reading
– Andreoni, J. ‘Impure altruism and donations to public goods: a theory of
warm-glow giving’, Economic Journal (1990) 100: 464—477.
– Abrams, B.A. and M.A Schmitz ‘The crowding out effect of government
transfers on private charitable contributions: cross sectional evidence’,
National Tax Journal (1984) 37: 563—568.
– Bergstrom, T.C., L. Blume, L. and H. Varian ‘On the private provision of
public goods’, Journal of Public Economics (1986) 29: 25—49.
– Bohm, P. ‘Estimating demand for public goods: an experiment’,
European Economic Review (1972), 3: 55—66.
– Cornes, R.C. and T. Sandler The Theory of Externalities, Public Goods
and Club Goods. (Cambridge: Cambridge University Press, 1996) [ISBN
0521477182 hcr] Chapters 6–10.

3. Reading
– Cullis, J. and P. Jones Public Finance and Public Choice, (Oxford:
Oxford University Press, 1998) [ISBN 0198775792 pbk] Chapter 3.
– Isaac, R.M., K.F. McCue and C.R Plott ‘Public goods in an experimental
environment’, Journal of Public Economics (1985), 26: 51—74.
– Itaya, J.-I., D. de Meza and G.D. Myles ‘In praise of inequality: public
good provision and income distribution’, Economics Letters (1997), 57:
289—296.
– Oakland, W.H. ‘Theory of public goods’ in A.J. Auerbach and M.
Feldstein (eds.), Handbook of Public Economics (Amsterdam: North-
Holland, 1987) [ISBN 044487612X hbk].
– Samuelson, P.A. ‘The pure theory of public expenditure’, Review of
Economics and Statistics (1954) 36: 387—389.
– Warr, P.G. ‘The private provision of a pure public good is independent of
the distribution of income’, Economics Letters (1983) 13: 207—211.
• Challenging reading
– Groves, T. and J. Ledyard ‘Optimal allocation of public goods: a solution
to the ‘free rider’ problem’, Econometrica (1977) 45: 783—809.

4. Reading
– Itaya, J.-I., D. de Meza and G.D. Myles ‘Income distribution, taxation
and the private provision of public goods’, Journal of Public Economic
Theory (2002) 4: 273—297.
– Foley, D.K. ‘Lindahl’s solution and the core of an economy with public
goods’, Econometrica (1970) 38: 66—72.
– Laffont, J.-J. ‘Incentives and the allocation of public goods’, in A.J.
Auerbach and M. Feldstein (eds.), Handbook of Public Economics.
(Amsterdam: North-Holland, 1987) [ISBN 044487612X hbk].
– Milleron, J.-C. ‘Theory of value with public goods: a survey article’,
Journal of Economic Theory (1972) 5: 419—477.

5. Introduction
• National defense: all inhabitants are
simultaneously protected
• Radio broadcast: received simultaneously
by all listeners in range of the transmitter
• These are both public goods
• If many consumers benefit from a single
unit of provision the efficiency theorems do
not apply

6. Definitions
• A pure public good satisfies:
– Nonexcludability If the public good is supplied,
no consumer can be excluded from
consuming it
– Nonrivalry Consumption of the public good by
one consumer does not reduce the quantity
available for consumption by any other
• A private good is excludable at no cost
and is perfectly rivalrous

7. Definitions
• Goods can possess
different combinations of
rivalry and excludability
• Club goods are studied in
chapter 6
• Common property
resources are studied in
chapter 7
• These are both examples
of impure public goods
Rivalrous
Non-
Rivalrous
Excludable
Non-
Excludable
Private
Good
Common
Property
Resource
Club
Good
Public
Good
Figure 5.1: Typology of goods

8. Private Provision
• Each consumer has an incentive to rely on
others to provide the public good
• The reliance on others is called free-riding
• This leads to inefficiency since too little
public good is provided
• All consumers will benefit from providing
more public good

9. Private Provision
• Consider two consumers who allocate
their incomes between a private good and
a public good
• The consumers take prices as fixed
• Each consumer derives a benefit from the
provision of the other
• This introduces strategic interaction into
the decision processes
• The Nash equilibrium has to be found

10. Private provision
• This interaction is captured in the preferences of
consumer with utility function
• The choice must satisfy the budget constraint
• Consider consumer 1, using the budget
constraint, his utility function is
h
)
,
( 2
1
g
g
x
U h
h

h
h
h
g
x
M 

)
,
( 2
1
1
1
1
g
g
g
M
U 


11. Private provision
• Indifference curves obtain by tracing the rate at which
can be traded for keeping utility constant
• Let be the best response of consumer 1 to
2
g
1
g
1
1
1
1
2
1 G
G
x
U
U
U
U
dg
dg 

2
g
1
1
G
x U
U 
1
1
ĝ
g 
1
1
G
x U
U 
1
1
ĝ
g 
1
1
G
x U
U 
1
1
ĝ
g 
1
ĝ

12. Private Provision
• Let be the provision of
consumer h
• Fig. 5.1 shows the
preferences of consumer
1
• Assume consumer 2
provides
• The utility of consumer 1
is maximized at
• Varying traces out the
locus of choices for
consumer 1
h
g
1
g
2
g
2
g
Budget
Constraint
1
ĝ
2
g
1
ĝ
2
g
Figure 5.2: Preferences and choice

13. Private Provision
• Fig. 5.2 constructs the
locus of choice for
consumer 2
• If consumer 1 chooses to
provide consumer 2
chooses
• The locus of chooses is
given by the solid line
• This is the best-response
function
2
ĝ
1
g
2
g
Budget
Constraint
1
g
2
ĝ
1
g
Figure 5.3: Best reaction for 2

14. Private Provision
• The Nash equilibrium is
where the choices of the
two consumers are the
best reactions to each
other
• Neither has an incentive
to change their choice
• This occurs at a point
where the best-response
functions cross
• The equilibrium choices
are and 2
ĝ 1
g
2
g
2
ĝ
1
ĝ
Figure 5.4: Nash equilibrium
1
ĝ

15. Private Provision
• The private provision
equilibrium is inefficient
• But it is privately rational
• A simultaneous increase
in provision by both
consumers gives a
Pareto improvement
• Pareto-efficient
allocations are points of
tangency between
indifference curves
Locus of Pareto
Efficient Allocations
Set of Pareto-
Improvements
1
g
2
g
2
ĝ
1
ĝ
Figure 5.5: Inefficiency of equilibrium

16. Efficient Provision
• At a Pareto-efficient allocation the indifference
curves are tangential
• This does not imply equality of the marginal
rates of substitution because the indifference
curves are defined over quantities of the public
good purchased by the two consumers
• Instead the efficiency condition involves the sum
of marginal rates of substitution and is termed
the Samuelson rule

17. Efficient Provision
• The tangency condition is
• Calculating the derivatives
• The marginal rate of substitution is
.
1
2
.
1
2
2
1 |
| const
U
const
U
dg
dg
dg
dg

2
2
2
1
1
1
G
x
G
G
G
x
U
U
U
U
U
U



h
x
h
G
h
x
G
U
U
MRS 
,

18. Efficient Provision
• The tangency condition then becomes
• This is the Samuelson rule
– The sum of marginal rates of substitution is equated
to the marginal rate of transformation between public
and private goods
– The marginal rate of substitution measures the
marginal benefit to a consumer of another unit of
public good
– The marginal rate of transformation is the marginal
cost of another unit
1
2
,
1
, 
 x
G
x
G MRS
MRS

19. Efficient Provision
• For two private goods the efficiency condition is
• Why the difference?
– An additional unit of a private good goes to either
consumer 1 or consumer 2
– Efficiency is achieved when both place the same
marginal value upon it
– An additional unit of public good benefits both
consumers
– The marginal benefits are therefore summed
2
,
1
, j
i
j
i MRS
MRS 

20. Allocation through Voting
• The level of public good provision is
frequently determined by voting
• Political parties promise different levels of
provision
• Majority voting determines which party
wins
• Need to assess whether this attains
efficiency

21. Allocation through Voting
• There is a population of H voters
• The cost of the public good is shared equally
• Consumer h has income
• The utility function is
• Each consumer votes for the value of G that
maximizes utility
h
M
  






 G
H
G
M
U
G
x
U h
h
,
,

22. Private
good
Public
good
Public
good
Utility
1
G H
G
m
G
Allocation through Voting
• Rank the consumers by
income so
• Fig. 5.5 shows that the
preferred levels of public
good satisfy
• Assume an odd number
of voters
• The median voter will be
decisive
• Their choice will win
the vote
H
M
M
M 

 ....
2
1
Figure 5.6: Allocation through voting
H
G
G
G 

 ....
2
1
m
G

23. Allocation through Voting
• The choice of the median voter satisfies
• The necessary condition can be written as
• So voting achieves efficiency only if
 






 G
H
G
M
U m
m
G
,
max
H
MRS m 1




H
h
h
m
H
MRS
MRS
1

24. Allocation through Voting
• Can any prediction be made?
– Income has a long right tail
– If MRS falls with income the median MRS is
greater than mean
– This implies voting results in Gm exceeding the
efficient level
• There is no guarantee that voting will
achieve efficiency

25. Personalized Prices
• With private goods
consumption is adjusted
to equate marginal
valuation with market
price
• With public goods it is not
possible for consumers to
adjust consumption
• This suggests adjusting
prices to match the
valuations of the fixed
quantity
• This is the basis of
personalized pricing
Private
good
Public
good
Price Same Different
Quantity Different Same
Table 5.1: Prices and quantities

26. Personalized Pricing
• Personalized pricing can be achieved by setting
the share of the public good financed by each
consumer
• The Lindahl mechanism asks each consumer to
announce public good demand as a function of
share
• The shares are adjusted until all consumers
demand the same quantity
• If the demands honestly reflect preferences the
equilibrium is efficient

27. Personalized Pricing
• The tax shares for the
two consumers are t1 and
t2
• The shares satisfy t1 + t2
= 1
• The budget constraint of
h is xh + thGh = Mh
• Gh is chosen to maximize
utility
• Equilibrium shares
ensure G1 = G2= G*
• Efficiency is achieved
1
t 2
t
2
G
1
G
*
G
*
G
Reaction
of 1
Reaction
of 2
Figure 5.7: Lindahl equilibrium

28. Personalized Pricing
• The choice problem is
• This has necessary condition
• Summing over consumers
• The allocation satisfies the Samuelson rule
   
h
h
h
h
h
G
G
G
M
U
h ,
max t

h
h
x
h
G
U
U
t

1
2
1
2
2
1
1



 t
t
x
G
x
G
U
U
U
U

29. Personalized Pricing
• Personalized pricing suffers from two
significant drawbacks
– There are practical difficulties of
implementation when there are many
consumers
– The Lindahl mechanism is not incentive
compatible and consumers have an incentive
to announce false demand functions

30. Personalized Pricing
• Assume consumer 1 is
honest
• If consumer 2 were also
honest the equilibrium
would be eL
• The equilibrium can be
moved to eM if consumer
2 announces a false
demand function
• Allocation eM maximizes
the utility of consumer 2
given the demand
function of consumer 1
1
t 2
t
2
G
1
G
L
e
M
e
Figure 5.8: Gaining by
false announcement

31. Mechanism Design
• Consumers will make false
announcements if this is advantageous
• This will distort the outcome
• Mechanism design is the search for
allocation mechanisms that cannot be
manipulated
• A preference revelation mechanism
ensures true preferences are revealed

32. Mechanism Design
• Understatement
– The benefit of the public
good is vh = 1
– Cost of 1 is met by those
reporting rh = 1
– Announcements either rh =
0 or rh = 1
– Provided if r1 + r2 ≥ 1
– Nash equilibrium is rh = 0, h
= 1, 2
– No provision: False
negative
Player 1
Player 2
0
0
1
1
0
0
0
0
1
1
2
1 2
1
Figure 5.9: Announcements
and payoffs

33. Mechanism Design
• Overstatement
– Benefit of the public
good is v1 = 0, v2 = ¾
– Cost of 1 is shared
equally
– Reports are r1 = 0 or 1,
r2 = ¾ or 1
– Provide if r1 + r2 ≥ 1
– Equilibrium r1 = 0, r2 =
1
– Inefficient provision:
False positive
Figure 5.10: Payoffs and
overstatement
Announcement
of Player 1
Announcement
of Player 2
0
1
1
0
0
4
1
2
1

2
1

2
1

4
1
4
1
4
3

34. Mechanism Design
• The Clarke-Groves mechanism ensures
– True values are revealed
– The public good is provided only when it
should be
• The allocation of cost is taken as given
• Consumers report their net benefits
(benefit – cost)
• Public good is provided if sum of net
benefits is positive

35. Mechanism Design
• If the public good is provided side
payments are made
• These side payments reflect the fact that
extracting the truth is costly
• The side payments internalize the net
benefit of the public good to other players

36. Mechanism Design
• Net benefits are vh = -1 or
vh = 1 (the mechanism
must work for both)
• Reports are rh = -1 or rh =
1
• The public good is
provided if r1 + r2 ≥ 0
• If provided the payoffs
are v1 + r2 for player 1 and
v2 + r1 for player 2
• The rh in the payoffs are
the side payments
Player 1
Player 2
-1
-1
+1
+1
0
0
1
1

v
1
1

v
1
1

v
1
2

v
1
2

v 1
2

v
Figure 5.11: Clarke-Groves
Mechanism

37. Mechanism Design
Player 1
Player 2
-1
-1
+1
+1
0
2

Player 1
Player 2
-1
-1
+1
+1
0 2
0
0
1
1


v 1
1


v
0 2
Figure 5.12: Payoffs for Player 1
• When v1 = -1 a truthful report is weakly dominant for player 1
• When v1 = 1 player 1 is indifferent between truth and false
statement
• The mechanism ensures there is no incentive not to be
truthful

38. Mechanism Design
• The mechanism ensures truthful reports
and efficient provision of the public good
• The drawback is the cost of the side
payments
• If v1 = v2 =1 the total cost of the side
payments is 2
• The side payments must be financed from
outside the mechanism

39. More on Private Provision
• The private provision model predicts
inefficiency
• The model also makes additional
predictions that can be contrasted to
evidence
• These predictions also have implications
for government policy

40. More on Private Provision
• Consider a transfer of D
from consumer 1 to
consumer 2
• The transfer shifts the
indifference curves from
the solid to the dashed
• The point
delivers the same utilities
after the transfer as the
point did before
• The best response
function also shifts
1
g
2
g
1
ĝ
D

1
ĝ
2
ĝ
D

2
ĝ
Figure 5.13: Effect of income
transfer
 
2
1
ˆ
,
ˆ g
g
 
D

D
 2
1
ˆ
,
ˆ g
g

41. More on Private Provision
• Consumer 1 reduces
contribution to the public
good by D
• Consumer 2 raises
contribution by D
• Total public good
provision remains at
• Consumption of private
good does not change
• The equilibrium is
invariant to the transfer
1
g
2
g
1
ĝ
D

1
ĝ
2
ĝ
D

2
ĝ
Figure 5.14: New equilibrium
2
1
ˆ
ˆ g
g
G 


42. More on Private Provision
• Assume H identical
consumers
• Let be provision of all
consumers but one
• Symmetry of equilibrium
implies
• As H increases the
equilibrium moves up the
reaction function
• The contribution of each
individual tends to zero
G
g
2

H
3

H
Pareto-
Efficient
Reaction
Function
Figure 5.15: Additional consumers
G
1


H
G
g

43. More on Private Provision
• The predictions of the model have been
tested using experiments
• The typical experiment gives participants a
fixed income to spend
• Income can be divided between a public
good and a private good
• The private good has higher private
benefit and the public good a higher social
benefit

44. More on Private Provision
• Fig. 5.16 shows typical
payoffs
• It is individually rational to
spend all income on the
private good
• It is socially optimal to
spend all income on the
public good
• The Nash equilibrium of
the one-shot game has
no investment in the
public good
Social Benefit
Private Benefit
Public Good
Private Good
5
5
1
10
Figure 5.16: Public good experiment

45. More on Private Provision
• In experiments average contribution to the
public good is 30 to 90 percent of income
• Most observations fall in the 40 to 50
percent range
• Among students the contribution to the
public good is lowest for economists and
falls with number of year of economics
• Repeating the game results in lower
contributions in later rounds

46. More on Private Provision
• Explanations:
• Fairness
• Warm glow
• Inability to play game
• Tradition

47. Contribution Campaign
• Explain what this is about

48. Fund-Raising Campaigns
• Describe diagram and
result
Time
Total
Contributions
...
T
x
2

T
x
4

T
x
3

T
y
1

T
y
C
T
Figure 5.17: A contribution campaign